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DTP/91-37—SPhT/91/108 [revised]

arXiv:hep-th/9109056v1 27 Sep 1991DTP/91-37—SPhT/91/108 [revised]A representation of the exchange relation foraffine Toda field theoryE. CorriganDepartment of Mathematical Sciences,University of Durham, Durham DH1 3LE, UKP.E.

DoreyService de Physique Th´eorique de Saclay,191191 Gif-sur-Yvette cedex, FranceVertex operators are constructed providing representations of the exchange relationscontaining either the S-matrix of a real coupling (simply-laced) affine Toda field theory, orits minimal counterpart. One feature of the construction is that the bootstrap relationsfor the S-matrices follow automatically from those for the conserved quantities, via analgebraic interpretation of the fusing of two particles to form a single bound state.September 19911 Laboratoire de la Direction des Sciences de la Mati`ere du Commissariat `a l’Energie Atomique

1. IntroductionStimulated by a study of perturbed conformal field theory [1,2], there has been some-thing of a revival of interest in two-dimensional affine Toda field theories, whose studywas begun long ago [3] but is nowhere near complete.

One of the striking features of two-dimensional integrable theories is the possibility of making plausible guesses for their exactS-matrices on the basis of the bootstrap and Yang-Baxter equations [4,5]. For the real cou-pling affine Toda field theories, based on the ADE Lie algebras, the Yang-Baxter equationitself plays no rˆole because the S-matrices are entirely diagonal.

Nevertheless, the proposedS-matrices enjoy an interesting analytic structure as a consequence of the bootstrap alone[6–13]. At least some of this interesting structure can be seen in perturbation theory butproofs of the conjectures are not yet available.

The same bootstrap structure appears inperturbations of certain conformal field theories, namely the coset models g(1) × g(1)/g(2).However, there the S-matrices are slightly different; while the affine Toda S-matrices havea factor with coupling constant dependent zeroes in the physical strip, this is absent fromthe proposals for perturbed conformal field theory. In fact, it is these ‘minimal’ S-matrices,of interest in their own right, that will be discussed first below.In [14,15], the bootstrap and its accompanying fusing rules were found to be intimatelyrelated to the geometry of root systems.

Moreover, formulae for the conjectured S-matriceswere discovered which made clearer their structural relationship with the roots and withthe bootstrap.The ideas might well have more general significance given the markedsimilarities between the affine Toda theory S-matrices and those conjectured, for example,for the principal chiral models [16]. Very similar mathematical structures have alreadybeen observed in the context of certain N = 2 Landau-Ginzburg models [17].The S-matrices for ADE affine Toda field theory can be written in a number of equiv-alent ways.

The expressions most suitable for the present discussion will be summarisedin section two, alongside some useful facts concerning the action of the Coxeter element ofthe Weyl group on the roots and weights. For more details concerning the latter, see forexample [18].Section three returns to an old idea in which the S-matrix appears in a ‘braiding’, or‘exchange’ relation [5,19]:Va(θa)Vb(θb) = Sab(θa −θb)Vb(θb)Va(θa),(1.1)where each of the operators V (θ) is formally associated with a particle of the theory, θdenoting its rapidity.1

According to the present understanding of real coupling affine Toda theory, eachoperator is a singlet since the particles are distinguished by conserved charges of non-zerospin. The assumption of associativity for the exchange relation then has no consequencefor the S-matrix—which is merely a set of numbers, one for each pair of particles.

In moregeneral situations, at least some particles will be degenerate and the associativity of (1.1)implies the Yang-Baxter or factorisation equation for the S-matrix.The expressions for the S-matrices given in section two are very suggestive and insection three a vertex operator representation of the exchange relation will be presented,thus providing a set of generating relations for the S-matrix in a fairly natural way. Thevertex operators to be used in this context are reminiscent of those used by Lepowsky andWilson [20,21] to obtain (twisted) representations of Kac-Moody algebras.

It appears atfirst sight that the twisted vertex operators are more appropriate in the context of affineToda field theory than those used by Frenkel, Kac and Segal [22] to provide the level onerepresentations of simply-laced affine algebras. Despite the similarities, there are somecrucial differences too, the most important being the absence of an obvious action of theconformal, or Virasoro, generators.

Since the field theories under discussion are massive,and therefore not conformal, this is hardly surprising.Given the exchange relation (1.1), it is natural to ask about the bootstrap itself.Suppose there is an ‘operator product’ expansion of Va(θa)Vb(θb), not in the sense of ashort distance expansion but rather in the sense of a bound-state fusing relation. In otherwords, suppose that, for certain (imaginary) rapidity differences, the two-particle statecontaining particles a and b is indistinguishable in terms of its quantum numbers fromanother (on-shell) single particle state c. From the point of view of the operators, it mightthen be expected that there should be a relation of the form,Va(θa)Vb(θb) ∼cabcVc(θc)(θa −θb −iU cab)nabc ,(1.2)whereθa ∼θc −iUbacθb ∼θc + iUabc,and U = π −U.

Momentum conservation (corresponding to the first conserved charge),together with the fact that the particles created by the operators V (θ) are always on-shell,requires that the relative rapidity of a and b is just i times U cab, the fusing angle for abound state in ab →ab scattering. It is a feature of the construction presented in section2

three that the numerical factor in (1.2) is meromorphic, the quantities nabc being certainintegers whose properties are described briefly at the end of that section.Multiplying (1.2) by Vd(θd) and using (1.1) leads to the bootstrap relation betweendifferent S-matrix elements:Scd(θc −θd) = Sad(θc −θd −iUbac)Sbd(θc −θd + iUabc). (1.3)The affine Toda field theories have infinitely many conserved charges Pr, P−r where, mod-ulo the Coxeter number h, the (positive) spin-label r runs over the exponents of the algebradefining the theory.

It would be expected that[Pr, Va(θa)] = parerθaVa(θa),(1.4)where par is the eigenvalue of the charge with spin r on the single type-a particle statePs+kh|pa >= pas+khe(s+kh)θa|pa > . (1.5)Given (1.4), (1.3) implies the bootstrap equation for the charges, namelypare−irUbac + pbreirUabc = pcr.

(1.6)In this article, a formalism will be developed far enough to provide a representationof (1.1) and (1.2) appropriate to the known minimal solutions of (1.3) and, after a simplemodification, to real coupling affine Toda S-matrices. Despite the absence of any derivationof these operators from a quantisation of the original affine Toda Lagrangian, their formis suggestive, and they seem to provide a natural setting in which to place the fusing ruleand S-matrix formulae described in [14,15].2.

PreliminariesThe discussion will be restricted to the theories associated with the simply-laced(ADE) series of Lie algebras. For a given theory, each particle is unambiguously associ-ated to one of the spots of the relevant Dynkin diagram.

This follows from the observation[7,23] (now proved Lie algebraically [24]) that the set of classical masses of the Toda par-ticles can be arranged to be the components of the smallest eigenvalue eigenvector of thecorresponding (non-affine) Cartan matrix. Besides picking a basis of simple roots for thealgebra, it also appears to be useful to divide the particles of the field theory into two sets.3

This division reflects a special property of root systems which allows the simple roots tobe split into two sets (black and white) so that within each set, all the roots are orthogonalto one another. The sets, the roots belonging to them, the particles themselves, and thefundamental weights associated with the simple roots in each set will be distinguishedwherever necessary by the symbols • or ◦.

(In ref [14] a slightly different notation wasused, the black roots being referred to as type α the white roots as type β. )Corresponding to the colouring of the simple roots, it is natural to choose a particularCoxeter element of the Weyl group.

Let wi be the Weyl reflection corresponding to thesimple root αi (i = 1, . .

., r) and definew• =Yi∈•wiw◦=Yi∈◦wiwhere the black set is labelled i = 1, . .

., b and the white set i = b + 1, . .

., r. Then w,defined byw = w•w◦,is a Coxeter element. Another subset of linearly independent roots is defined in terms ofthe set of simple roots αi in the following way.

Setφi = wrwr−1 . .

. wi+1αi(2.1)so that with the labelling introduced aboveφ◦= α◦andφ• = w◦α•.Moreover, if λi denotes the fundamental weights, satisfying (for simply-laced algebras),λi · αj = δij,then the roots defined by (2.1) satisfyφi = (1 −w−1)λi.

(2.2)This formula has also been used to relate the fusing rule given in [14] to the previouslyobserved Clebsch-Gordan property of the affine Toda couplings [25].4

It will be useful to have a (complex) basis of eigenvectors of the Coxeter element w.The elements of this basis are conveniently labelled by the exponents of the algebra. Thus,for each exponent s, there is an eigenvector es satisfyingwes = e2πis/hes(2.3)and normalised so thates · es′ = δs+s′,h(2.4)where h is the Coxeter number (ie order of the Coxeter element).

Occasionally, exponentsh/2 are repeated (in the Deven series). However, even in these cases the same notation willsuffice without confusion.The exponents of the algebra also label the eigenvectors of the Cartan matrix: foreach exponent there is an eigenvectorCijq(s)j= (2 −2 cos πs/h)q(s)i(2.5)and the eigenvectors are orthogonal for the ADE series.For computational purposes, it is often useful to have an expression for the basis (2.3)which makes its relationship with the eigenvectors of the Cartan matrix explicit.

For eachexponent s, definea(s)•=X•q(s)iαia(s)◦=X◦q(s)iαil(s)•=X•q(s)iλil(s)◦=X◦q(s)iλi. (2.6)Then, provided the eigenvectors of the Cartan matrix have unit length, the vectors a•, a◦are unit vectors while |l•| = |l◦| = 1/2 sin θs, where θs = sπ/h, and they enjoy a numberof other properties, includinga(s)•· a(s)◦= −cos θs,l(s)•· l(s)◦=cos θs4 sin2 θsa(s)•· l(s)◦= 0 = a(s)•· l(s)◦,a(s)•· l(s)•= a(s)◦· l(s)◦= 12.

(2.7)The eigenvectors of the Coxeter element can then be written in terms of these; for example,a convenient choice ises = ρs(a(s)•+ eisπ/ha(s)◦),(2.8)5

where ρs = 1/√2 sin θs. The choice of normalisation and the conditioneh−s = e∗s,(2.9)requirea(h−s)•= a(s)•a(h−s)◦= −a(s)◦(2.10)andq(s)•= q(h−s)•q(s)◦= −q(h−s)◦.

(2.11)Armed with these facts, it is straightforward to obtain a representation of roots or weightsin this basis; for example, the fundamental weights have components given byλ(s)k=(ρsq(s)k ,if k ∈•ρsq(s)k eisπ/h,if k ∈◦. (2.12)The masses of the affine Toda theory are proportional to the components of q(1).Moreover, assuming the other classically conserved quantities are preserved in the quantumtheory and are compatible with the bootstrap (1.6) leads to the conclusion [13,14] thatthe single-particle states in the quantum theory are eigenstates of the quantum operatorsPs+kh, with eigenvalues related to the eigenvectors q(s) of the Cartan matrix.

Invarianceunder parity requires par = pa−r and the fact that all fusing angles appearing in (1.6)are multiples of π/h requires pas = pas+2kh where k is any integer. Together, these tworequirements imply:pas+2kh ∝q(s)apas+(2k+1)h ∝q(h−s)a.

(2.13)Hence, using (2.11) above,p•s+kh ∝q(s)•p◦s+kh ∝(−)kq(s)◦. (2.14)(Note, s will always be taken to lie in the range 1, .

. ., h.)For spin ±1, the conserved charges are the light-cone momentum components and theeigenvalues are just the masses.

The particles are distinguished from each other using theconserved quantities but the single particle states will be labelled by their momenta only,all other labels being suppressed for convenience of notation.For later use, a couple of alternative expressions for the S-matrices for the variousaffine Toda theories will be given, each of them equivalent (with the proviso noted below)6

to the expressions provided in [14]. Here, as there and in earlier works [7,8], the blocknotation will be adopted in which the basic element of any of the conjectured S-matricesis constructed from the element(x, Θ)+ = sinhΘ2 + iπx2h(2.15)where Θ is the rapidity difference for the process and x is an integer.

The Θ dependence ismade explicit here for reasons which will become apparent in the next section. The basicbuilding block itself is then defined to be{x, Θ}+ =(x −1, Θ)+(x + 1, Θ)+(x −1 + B, Θ)+(x + 1 −B, Θ)+(2.16)where the functionB(β) = 12πβ21 + β2/4πcontains the conjectured coupling constant dependence.

The first expression, given in [15],can be summarised as follows:Sab(Θ) =hYp=1{2p + 1, Θ}λa·w−pφb+Θ = θa −θb(2.17)at least provided the two particles a and b share the same colour. If the two particlescorrespond to different colours then the appropriate expression is (2.17) but the particlelabelled ◦has its rapidity effectively incremented by iπ/h.

In other words, the appropriateS-matrix elements are obtained by replacing the rapidity Θ byΘ◦• = Θ + iπ/horΘ•◦= Θ −iπ/h. (2.18)There are other, equivalent expressions in terms of the ‘unitary’ block{x, Θ}+/{−x, Θ}+,but they are not so useful here.

The corresponding minimal S-matrix is obtained from(2.17) by simply deleting any β dependent term in (2.16).These S-matrices are unitary, satisfy crossing requirements and fulfil the bootstrapconditions on the odd order poles. They are analytic in the rapidity difference Θ with β-independent poles on the physical strip (ImΘ ∈[0, π]).

The poles may be of quite a highorder but appear to be compatible with perturbation theory as far as has been checked [9].7

It will be convenient in the next section to consider a two (complex) dimensionalspace of rapidity variables, θ and θ, corresponding to a complexification of the light-conemomentum variables p±. A representation of the exchange relation will be constructedin this complex space in the first instance and subsequently restricted to a ‘physical’submanifold.

In part, this is motivated by an analogy with conformal field theory where thetwo variables z and z, on which all conformal fields depend, are often treated as independentcomplex coordinates with a restriction to the euclidean section z∗= z left to a late stage ofa calculation. In the present case, it is the mass-shell condition pp = p+p−= m2, or θ = −θwhich selects the physical submanifold.

Note, the full vertex is automatically an analyticfunction of θ on this submanifold, in contrast to the conformal field theory situation inwhich a typical (non-chiral) vertex operator is the product of a function of z and a functionof z∗on the euclidean section. A feature of this kind is clearly needed—the S-matrix isitself analytic in θ, while typically, correlation functions in conformal field theory are not.Consider the minimal S-matrix, in which the factors containing the coupling constantdependence are omitted.

There are a number of continuations of (2.17) offthe Θ = −Θsub-manifold. Here, just one will be given:Sminab (Θab, Θab) = Fab(Θab, Θab)Fba(Θba, Θba)(2.19)where,Fab(Θab, Θab) =Qhp=1(−2p, Θab)λa·w−pλb+Qhp=1(−2p, Θab)λa·w−p−1λb+,(2.20)at least when the two particles share the same colour.

When the colours are different,both θ◦and θ◦are shifted by −iπ/h. Note, for θ◦this is the opposite sign to the shiftappearing in the previous formula (2.18).

To check agreement between (2.19) and (2.17)(when Θab = −Θab), the following inner product identities are useful:λ◦· w−pλ′◦= λ′◦· w−pλ◦λ• · w−pλ′• = λ′• · w−pλ•λ◦· w−pλ• = λ• · w−p−1λ◦. (2.21)It should be noted that whereas (2.17) is a meromorphic function of Θab this is not generallytrue for the expressions occuring in (2.20); since the inner products of weights are notusually integers, these functions will have individually a complicated cut structure.8

3. Representing the exchange relationThe basic ingredients of the construction will be described first and then elaboratedto provide a representation of the exchange relation for the minimal S-matrix.For each fundamental weight λ define a string-like, rapidity dependent field as follows:Xλ(θ) =Xr=s+khhr e−rθλ(h−s)cr.

(3.1)In (3.1), the sum extends over all integers k and exponents s and the Fock space annihilationand creation operators satisfy the commutation relations[cr, cr′] = (r/h)δr+r′,0. (3.2)The field Xλ is periodic in θ with period 2πi and a Coxeter rotation of the weight isequivalent to a shift 2πi/h in θ.

In that sense, the field is ‘twisted’. There is a groundstate satisfyingcr|0 >= 0r > 0.The commutation relation between the annihilation part of a field Xλ+(θ) and thecreation part of another, similar, field Xλ′−(θ′) is given by:hXλ+(θ), Xλ′−(θ′)i=hXp=1(λ · w−pλ′)ln1 −wpeθ′−θ,(3.3)at least provided Reθ > Reθ′.

Define a vertex operator to be the normal-ordered exponen-tial of such a fieldV λ(θ) = : expXλ(θ) : ≡eXλ−(θ)eXλ+(θ). (3.4)Then, the product of two vertex operators can be normal-ordered producing an extra factordepending upon the rapidity difference:V λ(θ)V λ′(θ′) =hYp=1(1 −exp(θ′ −θ + 2πip/h))λ·w−pλ′: V λ(θ)V λ′(θ′) :,(3.5)provided the rapidities bear the above relation to each other.

The factor on the righthand side of (3.5) is strikingly similar to the factors appearing in the S-matrix formulae(2.19) and (2.20), but not the same. A similar calculation with the ordering of the twooperators reversed produces the same factor (via manipulations valid in the complementary9

region Reθ < Reθ′ and up to a possible phase factor independent of rapidity). Hence, acomparison of the two orderings after analytic continuation in rapidity provides a trivialexchange relation.The vertex operator introduced above is a conformal operator with respect to theVirasoro generators built from the c-Fock space operators:Ln = 12Xrcrcn−r.This observation explains the feature just described.

To obtain an exchange relation cor-responding to the massive affine Toda field theories the vertex operator will need to be‘delocalised’ and its conformal nature destroyed.A second remark concerns the commutation relation of a Fock space operator withthe vertex:cs+kh, V λ(θ)= λ(s) e(s+kh)θ V λ(θ). (3.6)Recalling the earlier observations (2.12) and (2.14), it is tempting to regard the annihilationoperators in the Fock space as the conserved charges of the affine Toda theory, and takethe vertex operators (suitably modified) to describe the single particle states.

Indeed, the−iπ/h shift in rapidity for a type ◦particle, bearing in mind (2.14) and (2.12), renders (3.6)compatible with this assumption for either of the two colours. This rather natural pointof view is the one adopted here, at least tentatively, bearing in mind that the relationshipbetween this representation of the conserved quantities and the usual one in terms ofthe fundamental fields of the theory is missing, and that there is certainly a subtlety tobe understood with regard to the hermiticity of the operators concerned; the conservedquantitities are given classically as real functionals of the fields and their derivatives.Since, as remarked earlier, there are two mutually commuting sets of conserved quan-tities of opposite spin there will have to be at least one extra set of Fock space operators.These are quite naturally associated with the variable θ which will, as mentioned above,be regarded as independent of θ for most computational purposes.

The conserved quanti-ties with the opposite spin nevertheless share the same eigenvalues on the particle statesapart from their rapidity dependence. Thus, if an extra set of Fock space operators cr isintroduced they and their associated vertex operator Vλ(θ) will be expected to satisfyhcs+kh, Vλ(θ)i= λ(s) e(s+kh)θ Vλ(θ),(3.7)10

at least for r > 0. Note, this expression is also compatible with the shift in θ for a type ◦particle.Introducing a second set of operators also provides the opportunity for some ‘delocali-sation’ in the following sense.

The conformal nature of the vertex operator can be destroyedby shifting the rapidity dependence in the annihilation part of the vertex (ie without up-setting (3.7)) relative to that in the creation part. Indeed, a shift of 2πi/h seems to beexactly what is required to produce the minimal S-matrix (2.17).

Thus, introducing thefield Xλ(θ), it is convenient to setVλ(θ) = eXλ−(θ)eXwλ+ (θ)(3.8)and straightforward to repeat the normal-ordering calculation, givingVλ(θ)Vλ′(θ′) =hYp=11 −exp(θ′ −θ + 2πip/h)λ·w−p−1λ′: Vλ(θ)Vλ′(θ′) :,(3.9)this time valid in the region Reθ > Reθ′. Unfortunately, this is not quite what is requiredfor (2.20), the exponent in (3.9) has the wrong sign.

To put that right, the signature ofthe barred Fock-space operators should be reversed:[cr, cr′] = −(r/h)δr+r′,0. (3.10)The two calculations (3.5) and (3.9) indicate that a vertex operator associated with aparticle of type • could beV a(θ, θ) = V λa(θ)Vλa(θ)(3.11)while that associated with a particle of type ◦is a similar expression but with θ and θeach shifted by −iπ/h.

The latter is required to match the slight difference in the formulaefor the S-matrices for the two types, mentioned earlier. Adopting the composite vertexoperator and putting together the reordering effects givesV a(θa, θa)V b(θb, θb) = Fab(Θab, Θab) : V a(θa, θa)V b(θb, θb) :(3.12)where Fab is defined in (2.20).

Repeating the calculation in the opposite order gives a sim-ilar expression to (3.12), but with labels a, b reversed. Assuming an analytic continuationinto a common region of complex rapidity and comparing the two reordering expressionsyields the exchange relation, (1.1) for the minimal part of the S-matrix.11

Having achieved a representation of the minimal S-matrix, the same ideas may beadapted to introduce the coupling constant dependence. One way to do so is to introduceanother pair of string-like fields Y λ(θ) and Yλ(θ), with corresponding sets of annihilationand creation operators dr, dr, together with corresponding vertex operators W λ(θ) andWλ(θ) defined byW λ(θ) =: eY λ−(θ)eY λ+ (θ−iπ(2−B)/h) :Wλ(θ) =: eYλ−(θ)eYwλ+ (θ−iπ(2+B)/h) : .

(3.13)Note, both constituent vertex operators are ‘delocalised’ this time by an amount dependentupon B(β). Note also, the commutation relations of the d, d Fock space operators havethe same signature as the c, c operators, respectively.The operator representing a particle of type • is now taken to beV a(θa, θa) = V λa(θa)Vλa(θa)W λa(θa)Wλa(θa).

(3.14)Effectively, the four sets of annihilation and creation operators can be combined into afour-dimensional vector in a space with a metric of signature (+ + −−).This fact isreminiscent of a comment by Ward [26] concerning the embedding of an affine Toda fieldtheory in a self-dual gauge theory. This construction is certainly ad hoc and there maybe other, more subtle, ways of achieving the same goal.

Nevertheless, the example givenestablishes that the exchange relation can be represented in principle.Returning to the vertex operators it is worth emphasising that the coefficients of theFock space creation operators in (3.1) are proportional to the eigenvalues of the conservedquantities. This follows, irrespective of the colour of the particles, from the relations (2.12)and (2.13), taking into account the relative rapidity shift −iπ/h for the type ◦particles.In other words, the coefficients in (3.1) may be taken to be the conserved charges, up to afactor that may depend upon β and the spin, but not on the particle type.

Thus for anyparticle a, regardless of type, the string-like field may be writtenXa(θ) =Xr=s+khhr e−rθπa−rcr,(3.15)whereπas+kh ∝pas+kh.This remark is very interesting in the context of the fusing relation (1.2), since it impliesthat the fusing relation for the operators follows from the bootstrap for the eigenvalues of12

the conserved charges. To see this, consider a normal-ordered product of vertex operatorsfor a pair of particles a and b, such as that appearing on the right hand side of (3.5),and evaluate it for the values of rapidity appropriate to a fusing process (1.2).

Thus, it isrequired to evaluate: V a(θc−iUbac)V b(θc + iUabc) :=: exp" Xr=s+khh e−rθcrπa−reirUbac + πb−re−irUabccr#: . (3.16)However, using the conserved quantity bootstrap (1.6), valid for any spin r, and remem-bering the π’s and p’s are proportional with a factor independent of the species of particle,the right hand side of equation (3.16) is precisely V c(θc), as desired.

The same calculationworks for all the other pieces of any vertex operator, since the ‘delocalisation’ is indepen-dent of the particle type, and hence for the complete operators V a(θa, θa) defined in (3.11)or (3.14).It remains to study the singularity structure of the normal-ordered operator product(3.12), at least when the particles are on-shell so that θ = −θ. Only brief comments willbe made here, restricted to the minimal case, a fuller analysis will be given elsewhere.Firstly, although the numerator and denominator on the right hand side of (2.20) havebranch cuts (since the inner products of weights are not usually integers), when θ = −θthese combine nicely to yield poles and zeroes.

Moreover, the poles and zeroes also appearin the S-matrix, unless they cancel in the exchange relation. There are just two types ofcancelling poles and zeroes; zeroes at Θab = 0 and poles at Θab = iπ, the latter occuringonly when b = a, the former for b = a.

** The coefficient of the pole at Θaa = iπ in the aaoperator product is precisely unity. Otherwise, the poles come with associated zeroes; inother words, Fab is a product of factors of the form(x, Θ)n+(−x, Θ)m+,where m = n or n ± 1.

If m = n, then the poles and zeroes are destined to be even orderpoles and zeroes in the exchange relation and ought not to participate in the bootstrap; if** Note, the vertex operator V ∗a(θa, θa), obtained from (3.14) by reversing the sign of the weightλa on the right hand side, is in a sense, the operator conjugate to V a(θa, θa); it satisfies the sameexchange relation, it has a commutator with the conserved quantities with the opposite sign to(3.6), and there is a simple pole at θa = θ′a in the operator product of V a(θa, θa)V ∗a(θ′a, θ′a)13

m = n+1, the pole indicates the presence of a forward-channel bound state and participatesin the bootstrap; if m = n −1 it does not. In other words, it is the algebraic sum of polesand zeroes that appears to be relevant for the fusing relation, not the mere existence ofpoles.

Except by examining the S-matrix, it is not clear why this should be so. For adeeper analysis of the S-matrix pole structure, including an explanation of some of theabove remarks, see ref[15].4.

DiscussionDespite the lack of any construction in terms of the elementary Toda fields, the ver-tex operators presented here do at least provide a succinct summary of the affine TodaS-matrices and elucidate further the algebraic structure of these theories. Moreover, given(1.2), the S-matrix expressions (2.17) automatically satisfy (1.3), a fact which otherwiseneeded to be checked separately.

A particularly nice feature of the full vertex operator,expressed in terms of fields (3.15), is its dependence (without distinction of colour), onthe eigenvalues of the conserved charges. These statements apply equally well to the fullS-matrix or its minimal part, the vertex representation of the former being admittedly themore ad hoc.

An important ingredient in the construction presented here is the delocal-isation of the vertex operators as a means to obtain the necessary breaking of conformalsymmetry. This is fairly arbitrary, though in fact reminiscent of a vertex operator con-struction of quantum affine algebras, due to Frenkel and Jing [27].At a technical level, the problem of performing the analytic continuations to makethe comparison between the two halves of the operator reordering calculations in (3.12) isquite delicate because of the branch points in the terms in (2.20); there may be rapidity-independent phases to be taken care of by the introduction of matrix factors in the vertexoperators (see, for example, [21]).

In fact, the construction presented here is ‘heterotic’,being asymmetrical between the terms depending upon Θ and Θ. Indeed, as remarkedearlier, the part of the exchange relation arising from (3.5) is merely a phase independentof Θ.

However, it plays an important rˆole at intermediate stages, ensuring that the polesin the fusing relation (1.2) have integer powers. This and the desire to have quantities torepresent all (positive and negative) spin conserved charges motivated the inclusion of anapparently redundant set of Fock space operators.

It is also possible they serve to eliminatethe need for ‘cocycle’ factors of the type mentioned above.14

Finally, it is worth remarking that the construction of classical soliton solutions interms of τ-functions [28] involved vertex operators of a similar type, although in the contextof affine Toda theory a complete discussion of solitons is not yet available [29]. It will beinteresting to see how the classical information is related to the real β scattering theory.15

AcknowledgementsWe are grateful to D. Bernard for bringing ref[27] to our attention, to F. Smirnov andR. Sasaki for useful comments and to the Research Institute for Mathematical Sciences,Kyoto University for a fruitful visit.

One of us (E.C.) wishes to thank the Institute ofTheoretical Physics at the University of Santa Barbara, and the organisers of the Workshopon Conformal Field Theory for their hospitality during the initial stages of this work.The other (P.E.D.) is grateful to the Royal Society for a Fellowship under the EuropeanScience Exchange Programme.

The research was supported in part by the National ScienceFoundation under grant No.PHY89-04035, supplemented by funds from the NationalAeronautics and Space Administration, at the University of California at Santa Barbara.16

References[1]A. B. Zamolodchikov, Int.

J. Mod. Phys.

A4 (1989) 4235;V. A. Fateev and A. B. Zamolodchikov, Int.

J. Mod. Phys.

A5 (1990) 1025.[2]T. Eguchi and S-K Yang, Phys.

Lett. B224 (1989) 373;T. J. Hollowood and P. Mansfield, Phys.

Lett. B226 (1989) 73.[3]A.

V. Mikhailov, M. A. Olshanetsky and A. M. Perelomov, Comm. Math.

Phys. 79(1981) 473;G. Wilson, Ergod.

Th. and Dynam.

Sys. 1 (1981) 361;D. I. Olive and N. Turok, Nucl.

Phys. B215 (1983) 470.[4]M.

Karowski, Nucl. Phys.

B153 (1979) 244.[5]A. B. Zamolodchikov and Al.

B. Zamolodchikov, Ann. Phys.

120 (1979) 253.[6]A. E. Arinshtein, V. A. Fateev and A.

B. Zamolodchikov, Phys. Lett.

B87 (1979) 389.[7]H. W. Braden, E. Corrigan, P. E. Dorey and R. Sasaki, Proceedings of the NATO Con-ference on Differential Geometric Methods in Theoretical Physics, Lake Tahoe, USA2-8 July 1989 (Plenum 1990).[8]H.

W. Braden, E. Corrigan, P. E. Dorey and R. Sasaki, Nucl. Phys.

B338 (1990) 689.[9]H. W. Braden, E. Corrigan, P. E. Dorey and R. Sasaki, Nucl.

Phys. B356 (1991) 469.[10]H.

W. Braden and R. Sasaki, Phys. Lett.

B255 (1991) 343.[11]P. Christe and G. Mussardo, Nucl.

Phys. B330 (1990) 465;P. Christe and G. Mussardo, Int.

J. Mod. Phys.

A5 (1990) 4581.[12]C. Destri and H. J. de Vega, Phys.

Lett. B233 (1989) 336.[13]T.

R. Klassen and E. Melzer, Nucl. Phys.

B338 (1990) 485.[14]P. E. Dorey, Nucl.

Phys. B358 (1991) 654.[15]P.

E. Dorey, ‘Root systems and purely elastic S-matrices II,’ preprint RIMS-787/Saclay-SPhT/91-140, August 1991.[16]E. Ogievetsky and P. Wiegmann, Phys.

Lett. B168 (1986) 360;N. J. MacKay, Nucl.

Phys. B356 (1991) 729.[17]W.

Lerche and N. P. Warner, Nucl. Phys.

B358 (1991) 571.[18]N. Bourbaki, Groupes et alg`ebres de Lie IV, V, VI, (Hermann, Paris 1968);R. Carter, Simple Groups of Lie Type, (Wiley 1972);B. Kostant, Am.

J. Math.

81 (1959) 973.[19]F.A. Smirnov, in Introduction to Quantum Groups and Integrable Massive Models ofQuantum Field Theory, Nankai Lectures in Mathematical Physics, Mo-Lin Ge andBao-Heng Zhao (eds), (World Scientific 1990).[20]J.

Lepowsky and R.L. Wilson, Commun.

Math. Phys.

62 (1978) 43.[21]J. Lepowsky, Proc.

Natl. Acad.

Sci. USA 82 (1985) 8295.[22]I.

B. Frenkel and V. G. Kac, Inv. Math.

62 (1980) 23;G. Segal, Comm. Math.

Phys. 80 (1981) 301.17

[23]P. G. O. Freund, T. Klassen and E. Melzer, Phys. Lett.

B229 (1989) 243.[24]M. D. Freeman, Phys.

Lett. B261 (1991) 57;A. Fring, H.C. Liao and D. Olive, ‘The mass spectrum and coupling in affine Todatheories’ IC/TP-90-91/25.[25]H.W.

Braden, ‘A note on affine Toda couplings’, Edinburgh preprint 91-01.[26]R.S. Ward, Springer Lecture Notes in Physics 280 (1987) 106.[27]I.B.

Frenkel and N. Jing, Proc. Nat.

Acad. Sci.

U.S.A. 85 (1988) 9373.[28]M. Jimbo and T. Miwa, Vertex Operators in Mathematics and Physics, eds J. Lep-owsky, S. Mandelstam and I.M.

Singer, MSRI # 3 (Springer-Verlag New York 1985).[29]D. I. Olive and N. Turok, Nucl.

Phys. B265 (1986) 469;D. I. Olive and N. Turok, unpublished manuscript;T.J. Hollowood, private communication.18

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